Seidel and Thomas introduced some years ago a notion of a spherical object in the derived category D(X) of a smooth projective variety X. We introduce a relative analogue of this notion by defining what does it mean for an object E of the derived category D(Z x X) of a fiber product of two schemes Z and X to be spherical over Z.
For objects of D(Z x X) which are orthogonal over Z (these are categorical equivalents of a subscheme of X fibered over Z) we show an object to be spherical over Z if and only if it possesses certain cohomological properties similar to those in the original definition by Seidel and Thomas. We then interpret this geometrically for the special case where our objects are actual flat subschemes of X flatly fibered over Y. This is a joint work with Rina Anno of UChicago."

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